Weak Galerkin finite element methods for \(\mathbf{H}(\mathrm{curl};\Omega)\) and \(\mathbf{H}(\mathrm{curl},\operatorname{div};\Omega)\)-elliptic problems. (English) Zbl 1538.65530
Summary: Weak Galerkin finite element methods (WG-FEMs) for \(\mathbf{H}(\mathrm{curl};\Omega)\) and \(\mathbf{H}(\mathrm{curl},\operatorname{div};\Omega)\)-elliptic problems are investigated in this paper. The WG method as applied to curl-curl and grad-div problems uses two operators: discrete weak curl and discrete weak divergence, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on the arbitrary shape of polyhedra and, at the same time, is parameter-free. The optimal order of convergence is established for the WG approximations in discrete \(H^1\) norm and \(L^2\) norm. In fact, theoretical convergence analysis holds under low regularity requirements of the analytical solution. Results of numerical experiments that corroborate the theoretical results are also presented.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
| 78A25 | Electromagnetic theory (general) |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
curl-curl and gard-div problem; weak Galerkin finite element methods; polygonal/polyhedral meshesReferences:
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